I'm a lawyer, but not an immigration lawyer, so this may be out of date, but the last time I looked this up, entering the country illegally was a crime, but entering legally (e.g., on a tourist or student visa) and staying illegally (overstaying your visa) rendered you civilly deportable but wasn't a crime.
What about accepting welfare monies, health care; etc which you would not be entitled unless you are a citizen?
You are correct.
Entry without inspection (smeaking across without presenting yourself to a border agent) is a crime.
Visa overstays (legal entry but then staying beyond visa limits) is a civil offense.
Fraudulent entry (fake identity, lying about status) is a criminal offense.